Chaos: The Science of Predictable Random Motion

Description

Based on only elementary mathematics, this engaging account of chaos theory bridges the gap between introductions for the layman and college-level texts. It develops the science of dynamics in terms of small time steps, describes the phenomenon of chaos through simple examples, and concludes with a close look at a homoclinic tangle, the mathematical monster at the heart of chaos. The presentation is enhanced by many figures, animations of chaotic motion (available on a companion CD), and biographical sketches of the pioneers of dynamics and chaos theory. To ensure accessibility to motivated high school students, care has been taken to explain advanced mathematical concepts simply, including exponentials and logarithms, probability, correlation, frequency analysis, fractals, and transfinite numbers. These tools help to resolve the intriguing paradox of motion that is predictable and yet random, while the final chapter explores the various ways chaos theory has been put to practical use. Preface Acknowledgements Contents Part I Introduction 1 Chaos everywhere 1.1 Tilt-A-Whirl 1.2 Digits of 1.3 Butterfly effect 1.4 Weather prediction 1.5 Inward spiral Part II Dynamics 2 Galileo Galilei—Birth of a new science 2.1 When will we get there? 2.2 Computer animation 2.3 Acceleration 2.4 Free-fall 2.5 Reconstructing the past 2.6 Projectile motion Further reading 3 Isaac Newton—Dynamics perfected 3.1 Equations of motion 3.2 Force laws 3.3 Calculus Further reading 4 Celestial mechanics—The clockwork universe 4.1 Ptolemy 4.2 Copernicus 4.3 Brahe and Kepler 4.4 Universal gravitation 4.5 Circular orbits 4.6 Elliptical orbits 4.7 Clockwork universe Further reading 5 The pendulum—Linear and nonlinear 5.1 Rotational motion 5.2 Torque 5.3 Pendulum dynamics 5.4 Quality factor 5.5 Pendulum clock 5.6 Frequency 5.7 Nonlinearity 5.8 Where’s the chaos? Further reading 6 Sychronization—The Josephson effect 6.1 Hysteresis 6.2 Multistability 6.3 Synchronization 6.4 Symmetry breaking 6.5 Josephson voltage standard Further reading Part III Random motion 7 Chaos forgets the past 7.1 Period doubling 7.2 Random rotation 7.3 Statistics 7.4 Correlation 7.5 Voltage-standard redux Further reading 8 Chaos takes a random walk 8.1 Probability 8.2 Quincunx 8.3 Pascal’s triangle 8.4 Diffusion 8.5 Chaotic walk 8.6 In search of true randomness Further reading 9 Chaos makes noise 9.1 Beethoven’s Fifth 9.2 Fourier 9.3 Frequency analysis 9.4 Music to the ear 9.5 White noise 9.6 Random or correlated? Further reading Part IV Sensitive motion 10 Edward Lorenz—Butterfly effect 10.1 Lorenz equations 10.2 Exponential growth 10.3 Exponential and logarithmic functions 10.4 Liapunov exponent 10.5 Exponential decay 10.6 Weather prediction Further reading 11 Chaos comes of age 11.1 Kinds of chaos 11.2 Maxwell 11.3 Poincar´e 11.4 Hadamard 11.5 Borel 11.6 Birkhoff 11.7 Chaos sleeps 11.8 Golden age of chaos 11.9 Ueda 11.10 What took so long? Further reading 12 Tilt-A-Whirl—Chaos at the amusement park 12.1 Sellner 12.2 Mathematical model 12.3 Dynamics 12.4 Liapunov exponent 12.5 Computational limit 12.6 Environmental perturbation 12.7 Long-lived chaotic transients Further reading 13 Billiard-ball chaos—Atomic disorder 13.1 Joule and energy 13.2 Carnot and reversibility 13.3 Clausius and entropy 13.4 Kinetic theory of gases 13.5 Boltzmann and entropy 13.6 Chaos and ergodicity 13.7 Stadium billiards 13.8 Time’s arrow 13.9 Atomic hypothesis Further reading 14 Iterated maps—Chaos made simple 14.1 Simple chaos 14.2 Liapunov exponent 14.3 Stretching and folding 14.4 Ulam and von Neumann—Random numbers 14.5 Chaos explodes 14.6 Shift map—Bare chaos 14.7 Origin of randomness 14.8 Mathemagic Further reading Part V Topology of motion 15 State space—Going with the flow 15.1 State space 15.2 Attracting point 15.3 Contracting flow 15.4 Basin of attraction 15.5 Saddle 15.6 Limit cycle 15.7 Poincar´e–Bendixson theorem Further reading 16 Strange attractor 16.1 Poincar´e section 16.2 Saddle orbit 16.3 Period doubling 16.4 Strange attractor 16.5 Chaotic flow 16.6 Stretching and folding Further reading 17 Fractal geometry 17.1 Mathematical monster 17.2 Hausdorff—Fractal dimension 17.3 Mandelbrot—Fractal defined 17.4 Physical fractals 17.5 Fractal attractor Further reading 18 Stephen Smale—Horseshoe map 18.1 Horseshoe map 18.2 Invariant set 18.3 Symbolic dynamics 18.4 3-D flow 18.5 Structural stability 18.6 Protests and prizes Further reading 19 Henri Poincar´e—Topological tangle 19.1 Homoclinic point 19.2 Homoclinic trajectory 19.3 Homoclinic tangle 19.4 Fixed-point theorem 19.5 Horseshoe 19.6 Poincar´e–Birkhoff–Smale theorem 19.7 Heteroclinic tangle 19.8 Fractal basin boundary 19.9 Robust chaos 19.10 Paradox lost 19.11 Stability of the Solar System Further reading Part VI Conclusion 20 Chaos goes to work 20.1 Randomness 20.2 Prediction 20.3 Suppressing chaos 20.4 Hitchhiker’s guide to state space 20.5 Space travel 20.6 Weather modification 20.7 Adaptation 20.8 Further reading Bibliography Index